The answer is no.
In a cancellative monoid left invertible elements are right invertible. If yx =1 then xyx=x and so xy =1 by cancellation. So you are asking to invert a given element in some cancellative monoid.
Take a fg cancellative monoid M not embeddable in a group. Invert the generators one by one to embed M in the group of units of a monoid and get a contradiction.